Some Special Types of Orbits Around Jupiter

aerospace

Article Some Special Types of Orbits around Jupiter

Yongjie Liu 1, Yu Jiang 1,*, Hengnian Li 1,2,* and Hui Zhang 1

1 State Key Laboratory of Astronautic Dynamics, Xi’an Satellite Control Center, Xi’an 710043, China; [email protected] (Y.L.); [email protected] (H.Z.) 2 School of Electronic and Information Engineering, Xi’an Jiaotong University, Xi’an 710049, China * Correspondence: [email protected] (Y.J.); [email protected] (H.L.)

Abstract: This paper intends to show some special types of orbits around Jupiter based on the mean element theory, including stationary orbits, sun-synchronous orbits, orbits at the critical inclination,

and repeating ground track orbits. A gravity model concerning only the perturbations of J2 and J4 terms is used here. Compared with special orbits around the Earth, the orbit dynamics differ greatly: (1) There do not exist longitude drifts on stationary orbits due to non-spherical gravity

since only J2 and J4 terms are taken into account in the gravity model. All points on stationary orbits are degenerate equilibrium points. Moreover, the satellite will oscillate in the radial and North-South directions after a sufﬁciently small perturbation of stationary orbits. (2) The inclinations of sun-synchronous orbits are always bigger than 90 degrees, but smaller than those for satellites around the Earth. (3) The critical inclinations are no-longer independent of the semi-major axis and eccentricity of the orbits. The results show that if the eccentricity is small, the critical inclinations will decrease as the altitudes of orbits increase; if the eccentricity is larger, the critical inclinations will increase as the altitudes of orbits increase. (4) The inclinations of repeating ground track orbits are monotonically increasing rapidly with respect to the altitudes of orbits.

Keywords: mean element theory; orbit dynamics; longitude drift; critical inclinations

Citation: Liu, Y.; Jiang, Y.; Li, H.; Zhang, H. Some Special Types of Orbits around Jupiter. Aerospace 2021, 8, 183. https://doi.org/10.3390/ 1. Introduction aerospace8070183 Jupiter is the most massive planet in the solar system. It is also a gas giant planet. Its mass is 2.5 times the mass of other planets in the solar system. The large size of Jupiter also Academic Editor: Fanghua Jiang makes it relatively easy to be observed. As a result, it was discovered very early. Jupiter has been one of the major targets for planetary exploration. However, Jupiter’s powerful Received: 19 April 2021 magnetosphere and radiation belts are threats to all human spacecraft trying to visit Jupiter. Accepted: 4 July 2021 Since the 1970s, several space missions have been launched by NASA, such as Pioneer X, Published: 8 July 2021 Voyager 1, Galileo, and Juno. However, there is still a long way to go to explore Jupiter. There exist many interesting phenomena that have attracted people to explore for many Publisher’s Note: MDPI stays neutral years, for example, the Great Red Spot, Jupiter’s rings, and atmospheric jet streams. As with regard to jurisdictional claims in a gaseous planet, Jupiter cannot be explored by landing like a lithospheric planet. We published maps and institutional affil- can only use probes to orbit and enter Jupiter’s atmosphere. Therefore, it is necessary to iations. investigate the orbit dynamics around Jupiter. Weibel et al. [1] researched stable orbits between Jupiter and the Sun. Jacobson [2] investigated the gravity field of Jupiter and the orbits of its Galilean satellites. Colwell et al. [3] studied the exogenic dust ring. Recently, Liu et al. [4,5] discussed the dust in the Jupiter system outside the rings and distribution of Copyright: © 2021 by the authors. Jovian dust ejected from the Galilean satellites. Research about this will surely contribute Licensee MDPI, Basel, Switzerland. to the orbit design of space missions. This article is an open access article Investigating the dynamical environments around planets has been the focus for distributed under the terms and space missions in the past few decades. For this purpose, much research concerning conditions of the Creative Commons various special artificial satellite orbits around planets have been conducted. Usually, these Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ special types of orbits include stationary orbits, frozen orbits, sun-synchronous orbits, 4.0/). repeating ground-track orbits, and orbits at the critical inclination. The original idea about

Aerospace 2021, 8, 183. https://doi.org/10.3390/aerospace8070183 https://www.mdpi.com/journal/aerospace Aerospace 2021, 8, 183 2 of 15

geostationary orbits was first put up by Clarke [6]. He pointed out that satellites with an altitude of 36,000 km above the equator of the Earth would have the same rotation rates as the Earth and stay stationary relative to an observer on the equator. Therefore, geostationary satellites are often used for the sake of communications and navigations. Four equilibrium solutions for geostationary orbits were shown to exist by Musen & Bailie [7]. Moreover, two of them were stable while the other two were unstable. Lara & Elipe [8] calculated periodic orbits around equilibrium points in the Earth second degree and order gravity field. For Mars, the stationary orbits, also known as areostationary orbits, and the equilibrium points were studied by Liu et al. [9]. The periodic orbits around the equilibrium points were also calculated by Liu et al. [10]. For orbits at the critical inclination, the eccentricity and argument of perigee are invariant on average. The concept of the Earth critical inclination was first introduced by Orlov [11]. Brouwer [12] used canonical transformations to eliminate short-period terms. Coffey et al. gave a geometrical interpretation of the critical inclination for satellites by investigating the averaged Hamilton system [8]. Representatives of orbits at the critical inclination are the Russian Molniya satellites. The combined effects of the critical inclination and the 2:1 mean motion resonance of a Molniya orbit have been intensively studied since then, for example, in [13–17]. Similarly, frozen orbits are characterized by the invariance of average eccentricity and argument of perigee. Frozen orbits are not limited to specific inclinations. They may exist at any inclination. Usually, the argument of perigee is equal to 90 or 270 deg, depending on the sign of the ratio of the harmonic coefficients J3 and J2. Frozen orbits were first proposed by Cutting et al. [18] for orbit analysis of the Earth satellite SEASAT-A. Coffey et al. [19] showed that there exist three families of frozen orbits in the averaged zonal problem up to J9 in the gravity field of an Earth-like planet. The frozen orbits around the moon in the full gravity model were considered by Folta & Quinn [20], and Nie & Gurfil [21]. Some researchers also view orbits at the critical inclination as frozen orbits, for example [19,22]. Sun-synchronous orbits are defined with a precession rate of the orbital plane equal to the revolution angular velocity around the sun. Generally, remote sensing satellites are placed into these orbits. Macdonald et al. [23] used an undefined, non-orientation- constrained, low-thrust propulsion system to consider an extension of the sun-synchronous orbits. For repeating ground track orbits, the trajectory ground track repeats after a whole number of revolutions within some days. Orbits of this type are widely used to achieve better coverage properties. Lara showed that orbits repeating their ground track on the surface of the Earth were members of periodic-orbit families of the tesseral problem of the Earth artificial satellite [24]. Lei [25] considered the leading terms of the Earth’s oblateness and the luni-solar gravitational perturbations to describe the secular dynamics of navigation satellites moving in the medium Earth orbit and geosynchronous orbit regions. Liu et al. [9] calculated these five types of special orbits around Mars with analytical formulations and numerical simulations. In fact, the gravity field of Mars shares many similarities with that of the Earth. The J2 terms of them are dominant among the harmonic coefficients. However, the J2 term is not as dominant as Earth’s J2. The other first few harmonic coefficients are also strong for Mars: about 1–2 orders of magnitude smaller than J2; for the Earth, the other first few harmonic coefficients are about 3–4 orders of magnitude smaller than J2. The situation is rather different for Jupiter compared with the Earth and Mars. Due to the difficulty of determining the gravity field of Jupiter, there exist few studies about special types of orbits around Jupiter, as far as we know. However, the situation has greatly improved since Juno’s gravitational measurements were conducted. Iess et al. [26] provided measurements of Jupiter’s gravity harmonics (both even and odd) through precise Doppler tracking of the Juno spacecraft. Moreover, they pointed out a North-South asymmetry, which is a signature of atmospheric and interior flows. Here, we mainly use the results Aerospace 2021, 8, 183 3 of 15

in [26] to build a simpliﬁed gravity model of Jupiter. Some harmonic coefﬁcients of the gravity model of the Earth, Mars, and Jupiter can be seen in Table1.

Table 1. Some harmonic coefﬁcients and their ratios of the Earth, Mars, and Jupiter.

−3 −6 −5 −6 −3 −2 Planet J2(×10 ) J3(×10 ) J4(×10 ) J22(×10 ) J3/J2(×10 ) J4/J2(×10 ) Earth ([27]) 1.08263 −2.53266 −1.61962 1.81534 −2.3394 −1.4960 Mars ([28]) 1.95545 31.4498 −1.53774 63.0692 16.083 −0.7864 Jupiter ([26]) 14.6965 0.042 −58.661 0 0.0021378 −3.9923

In this paper, we investigate some special orbits around Jupiter, considering mainly the effect of the non-spherical perturbation of the gravity ﬁeld. From Table1, one can see that the J2 term is still dominating, J2 is about 25 times the 5 value of J4, but is 10 times bigger than J3. The terms, such as J3, J21, J22, J5, J6 (the values and their uncertainty can be seen in [26]), can be neglected compared with the terms of J2 and J4. Therefore, a good approximation of the gravity model of Jupiter is given by " # µ J R 2 J R 4 U = 1 − 2 3 sin2 φ − 1 − 4 35 sin4 φ − 30 sin2 φ + 3 , (1) r 2 r 8 r

−11 3 −1 −2 where µ = GMJ is the gravitational constant of Jupiter, G = 6.67428 × 10 m · kg · s , MJ is the mass of Jupiter, R is the radius of Jupiter, r is the distance of the satellite relative to the center of mass of Jupiter, φ is the latitude of the satellite. Equation (1) indicates that the gravity model of Jupiter that we use here is symmetrical with respect to the z-axis. This leads to different characteristics of satellites orbiting around Jupiter and the Earth or Mars. In the next sections, we adopt the gravity model represented by Equation (1) and use it to study some mean features of orbits around Jupiter.

2. Stationary Orbits Satellites on stationary orbits are well-known for their stationary ground track. There- fore, stationary orbits are preferred for designing communications and navigation satellites. There exist numerous studies on stationary orbits of the Earth and Mars. However, the gravity ﬁeld of Jupiter is signiﬁcantly different. In this subsection, we will calculate the stationary orbit of Jupiter and investigate their stability in a spherical coordinate system. In the spherical coordinates of an inertial frame, O − r, λ, φ, where O is the center of mass of Jupiter, λ is the jovicentric longitude, r and φ are the same as those in Equation (11), and the kinetic energy of the spacecraft can be written as 1 . 2 . 2 . 2 T = r + r2 cos2 φλ + r2φ , (2) 2

. . . where r, λ, φ are the derivatives with respect to time. From the expressions of T and U, one can see that λ is a cyclic variable. Let us introduce q = [r, λ, φ]T, the Lagrangian can be written as L = T − U. By Lagrange equations, we have

d ∂T ∂T ∂U . − = . (3) dt ∂q ∂q ∂q

More precisely, the equations of motion can be presented as follows:  .. . 2 . 2 2 4 r − r cos2 φλ − rφ = − µ + 3µJ2R 3 sin2 φ − 1 + 5µJ4R 35 sin4 φ − 30 sin2 φ + 3 ,  r2 2r4 8r6  . d 2 2 dt r cos φλ = 0, (4)  . . 2 h 2 4 i  d r2 + 1 r2 ( ) = − 3µJ2R + µJ4R 2 − dt φ 2 sin 2φ λ 2r3 8r5 70 sin φ 30 sin 2φ. Aerospace 2021, 8, x FOR PEER REVIEW 4 of 17

 .. μ 35μμJR24 JR r−−=−+−+−+rrcos22φλ φ 224() 3sin 2 φ 1() 35sin 4 φ 30 sin 2 φ 3 ,  rr2428 r 6   d 22  ()r cos φλ = 0, (4) dt  d 1 3μμJR24 JR ()22φφλ+=−+−() 224() 2 φφ  rrsin270sin30s. 35  in 2 dt 2  28rr 

From the second equation of (4), we see that the quantity r 22cos φλ is invariant, which can also be obtained by conservation of the angular momentum along the z-axis. Moreover, the zonal terms of the gravity field only lead to radial and North-South perturbations. The presence of these terms increases the radius of the stationary orbit with respect to the case of a spherical planet with the same mass of Jupiter. When the orbital ⋅ plane coincides with the equatorial plane, namely φφ==0, 0 , the vertical perturbation ==λλφφ = = == vanishes. In order to find stationary orbits for Jupiter, let rr0, nJ , 0, 0 in these equations; we get  μ 315μμJ RJR24 n2 =+24 − ,  J rr3528 r 7 (5)  sin 2φ = 0, Aerospace 2021, 8, 183 4 of 15 or

24.  μ 35μμJR JR2 2 From the second equation−+ of (4),24 we see that24222 theφφφφ −+ quantity r cos φλ is invariant, − which ++ =  ()3sin 1() 35sin 30sin 3rncos J 0 , can also be obtained by conservationrr2428 of the angular momentum r along 6 the z-axis. Moreover, the zonal terms of the gravity ﬁeld only lead to radial and North-South perturbations. The (6) 1 3μμJR24 JR presence of these terms increasesrn22+= the radius24 of+− the stationary()70 sin orbit 2φ with 30 respect 0, to the case of a spherical planet with theJ same mass35 of Jupiter. When the orbital plane coincides with 2 28· rr the equatorial plane, namely φ = 0, φ = 0, the vertical perturbation vanishes. In order to ...... ﬁnd stationary orbits for Jupiter, let r = r = 0, λ = n , λ = 0, φ = φ = 0 in these equations; where nJ is the rotational angularJ velocity of Jupiter. One can verify that the left-hand we get ( 2 4 side of the second equationn2 = µ + in3µJ 2(6)R − is15 µalwaysJ4R , positive when r is larger than R . Therefore, J r3 2r5 8r7 (5) we only need to analyzesin 2 φsolution= 0, s of Equation (5). From the second equation of (5), we see or π φ = φ π that only one2 meaningful4 latitude of the stationary orbit exists, i.e., 0 ( =, are  − µ + 3µJ2R 3 sin2 φ − 1 + 5µJ4R 35 sin4 φ − 30 sin2 φ + 3 + r cos2 φn 2 = 0, r2 2r4 8r6 J 2 2 4 (6)  1 r2n 2 + 3µJ2R + µJ4R 70 sin2 φ − 30 = 0, π also2 rootsJ 2rof3 sin8r5 () 2φ = 0 , but φ= makes no sense for stationary orbits, and φ=π where nJ is the rotational angular velocity of Jupiter. One2 can verify that the left-hand side ofcorresponds the second equation to ina (6)stationary is always positive orbit when whichr is larger coincides than R. Therefore, with wethat only of φ=0 ). Introducing the func- need to analyze solutions of Equation (5). From the second equation of (5), we see that only one meaningful latitudeμ of theμμ stationary24 orbit exists, i.e., φ = 0 (φ = π , π are also rootsμ 315JR24 JR 2 2 2 of sin(s2φ) = 0, but=+φ = π makes no sense − for stationary − orbits, and φ = π correspondsf ()rn=− tion f0 ()rn352 7J and 1 3 J , the variations of to a stationary orbit whichrr coincides28 with that r of φ = 0). Introducing the functionsr 2 4 f (r) = µ + 3µJ2R − 15µJ4R − n2 and f (r) = µ − n2, the variations of f (r), f (r), with 0f01(),rfrr3 ()2r5 , with8r7 respectJ to1 r , rcan3 Jbe seen in the0 following1 Figure 1. respect to r, can be seen in the following Figure1.

Figure 1. Variations of f0(r), f1(r) as r increases from 2.2R to 2.3R. Figure 1. Variations of f01(),rfr () as r increases from 2.2R to 2.3R . It can be proved that f0(r) vanishes at two real positive values of r, but only one of 8 them is bigger than R. This solution, which is given by r0 = 2.2414R = 1.6024 × 10 m, is denoted by E0 in Figure1. If the perturbation due to the non-spherical gravity ﬁeld is not considered, one can ﬁnd that the radius for stationary orbits is about 2.2381R, which corresponds to the point E1. Therefore, the existence of the J2 and J4 terms increase the altitude of the stationary orbit. Aerospace 2021, 8, 183 5 of 15

To study the stability of the stationary orbits under small perturbations, we mainly used the epicyclic theory (Murray & Dermott [29]). We can denote by the constant h, the . value taken by r2 cos2 φλ for some given initial conditions. Assuming that the deviations from the stationary orbits are small and writing r = r0(1 + ε), Equation (4) can be linearized as follows (Murray & Dermott [29], Section 11 of Chapter 6):  ..  ε + (3α1 + α2)ε = 0,  .. φ + (α1 + β2)φ + β1 = 0, · (7)  − h =  λ 2 2 2 0 r0 (1+ε) cos φ

1 ∂U ∂2U 1 ∂U 1 ∂2U where α1 = − (r0, 0), α2 = − 2 (r0, 0), β1 = − 2 (r0, 0), and β2 = − 2 2 (r0, 0). r0 ∂r ∂r r0 ∂φ r0 ∂ φ By solving Equation (7), the analytical approximate expressions of ε, φ, λ can be formulated as [29]  ε = e cos k t,  1  β1 φ = − + i cos k2t, α1+β2 √ (8)  √ 2 α  λ = α t − 1 e sin k t, 1 k1 1 2 2 where k = 3α1 + α2, k = α1 + β2, i is the inclination, and e is the eccentricity of the orbit. 1 2 . √ √ Note that the average of λ is α1. Let us set k3 = α1. The ﬁrst two equations in (7) or (8) show that the radial and North-South motions are uncoupled. Using the expression (1), we can get a precise form of the three frequencies k1, k2, k3 [29]:

 2 4  k 2 = µ − 3 J R + 45 J R  1 r 3 1 2 2 r 8 4 r ,  0 0 0  2 4 2 µ 9 R 75 R k2 = 3 1 + 2 J2 r − 8 J4 r , (9)  r0 0 0  2 4  2 µ 3 R 15 R  k3 = 3 1 + J2 − J4 .  r0 2 r0 8 r0

Therefore, the satellites will oscillate in the radial and North-South directions with frequencies k1 and k2, respectively. Moreover, the mean motion frequency, k3, in the West- East direction is larger than in the Keplerian case. Namely, for a given semi-major axis, the satellite moves faster than the rate expected at that location in the Keplerian case. In the following, we give some numerical examples to illustrate the above characteristics. We computed the evolution of r, φ, and λ when we selected initial values of r and φ close to r0 and 0, respectively. Figure2 shows a stationary orbit and four orbits obtained from r = 0.99r0, φ = ±0.1 deg and r = 1.01r0, φ = ±0.1 deg. We can see that these four disturbed orbits are no longer periodic. This can be explained by using Equation (9). Due to the presence of the terms that contain J2 and J4, the three frequencies are usually not equal or even commensurable. Figure3 illustrates that the satellite oscillates in the radial and North-South directions. On the other hand, the satellite on stationary orbits does not drift . in these directions. It only moves with a constant λ along the orbit. Therefore, the drift of longitude would not occur for satellites on stationary orbits since only zonal harmonics are taken into account in the gravity model. Furthermore, there are no signiﬁcant differences between points on the stationary orbit. As a result of the symmetry of the gravity ﬁeld, it can be concluded that the points on stationary orbits are degenerate equilibrium points. Here, an equilibrium point is degenerate if, and only if, the matrix of the linearized equation for (4) is degenerate. This is the major difference with respect to stationary orbits of other planets, such as the Earth and Mars. For the Earth and Mars, there exist four equilibrium points on stationary orbits, among which two are stable and the other two are unstable. Aerospace 2021, 8, x FOR PEER REVIEW 6 of 17

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==±φ ==±φ obtained from rr0.990 , 0.1 deg and rr1.010 , 0.1 deg. We can see that these four disturbed orbits are no longer periodic. This can be explained by using Equation (9). ==±φ ==±φ obtained from rr0.990 , 0.1 deg and rr1.010 , 0.1 deg. We can see that these Due to the presence of the terms that contain J 2 and J 4 , the three frequencies are usu- allyfour not disturbed equal or orbits even arecommensurable. no longer periodic. Figure This 3 illustrates can be explained that the bysatellite using oscillatesEquation in(9). theDue radial to the and presence North-South of the directions. terms that On contain the ot herJ 2 hand,and J the4 , the satellite three on frequencies stationary are orbits usu- doesally not not drift equal in orthese even directions. commensurable. It only moves Figure with 3 illustrates a constant that λ the along satellite the orbit. oscillates There- in fore,the radialthe drift and of North-South longitude would directions. not occur On the fo rot satellitesher hand, on the stationary satellite on orbits stationary since onlyorbits zonaldoes harmonics not drift in are these taken directions. into account It only in moves the gravity with a model. constant Furthermore, λ along the there orbit. are There- no significantfore, the driftdifferences of longitude between would points not on occur the st foationaryr satellites orbit. on As stationary a result of orbits the symmetry since only ofzonal the gravity harmonics field, are it can taken be concludedinto account that in the the points gravity on model. stationary Furthermore, orbits are degeneratethere are no equilibriumsignificant points.differences Here, between an equilibrium points on point the st isationary degenerate orbit. if, As and a resultonly if, of the the matrixsymmetry of theof linearizedthe gravity equation field, it can for be(4) concluded is degenerate. that theThis points is the on major stationary difference orbits with are respectdegenerate to stationaryequilibrium orbits points. of other Here, planets, an equilibrium such as thepoint Earth is degenerate and Mars. if, For and the only Earth if, the and matrix Mars, of therethe linearizedexist four equilibriumequation for points (4) is degenerate.on stationary This orbits, is the among major which difference two arewith stable respect and to thestationary other two orbits are unstable. of other planets, such as the Earth and Mars. For the Earth and Mars, Aerospace 2021, 8, 183 6 of 15 there exist four equilibrium points on stationary orbits, among which two are stable and the other two are unstable.

− − = φ =− − Figure 2. Orbits around Jupiter: green stationary orbits; red rr0.99 0 , 0.1 deg; blue = φ = − = φ =− − = φ = rr0.99 0 and 0.1 deg; cyan rr1.01 0 and 0.1 deg; black rr1.01 0 and 0.1 Figure 2. Orbits around Jupiter: green—stationary− orbits; red—− r = 0.99r , φφ ==−−0.1 deg; blue—− deg.Figure 2. Orbits around Jupiter: green stationary orbits; red rr0.99 00, 0.1 deg; blue r ==0.99r0 and φφ==0.1 deg; cyan—r−= 1.01= r0 and φ =φ−=−0.1 deg; black—r −= 1.01= r0 and φ = 0.1φ = deg. rr0.99 0 and 0.1 deg; cyan rr1.01 0 and 0.1 deg; black rr1.01 0 and 0.1 deg.

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(a) (b)

(e) (f)

FigureFigure 3. 3.(a,b(a,)b Variations) Variations of of orbital orbital radius radius ( (rr),), ((cc,d,d)) driftdrift ofof longitudelongitude ( ∆( Δλλ), (),e ,(fe,f) evolution) evolution of of latitude lati- φ − = φ =− − = tude(φ): ( green—stationary): green stationary orbits; orbits; red, r =red,0.99 rrr0 and0.99φ0=and−0.1 deg;0.1 blue— deg;r =blue0.99 r0 andrr0.99φ = 00.1 anddeg; φ = = − = = − φ =− = =− = φ = cyan—0.1 deg;r 1.01cyanr0 andrrφ 1.010.10 and deg; black—0.1r deg;1.01 blackr0, φ 0.1rr deg,1.01 and0 , T is a0.1 Joviandeg, day.and T is a Jovian day.

In the following, we show that the effects of third bodies, the Jovian ring and the

magnetic field of Jupiter, are negligible compared to those of the J 2 and J 4 terms. Based on the data of the Sun and moons around Jupiter (see for example, [30]), one

can calculate the maximal ratio of the disturbed acceleration ( ad ) and central gravity ac-

celeration ( am ) for satellites on stationary orbits, which is achieved when the satellite, Ju- piter, and the third body are in a straight line. Therefore, the maximal ratio can be calculated as [31]

3 ard m 0 = 2 , (10) aMρ mJmax  where m is the mass of the third body, and ρ is the distance between Jupiter and the third body. Values of the maximum ratio (109) for different bodies are reported in Table

2. Since the J 4 term of Jupiter is 100 times bigger than the ratio of acceleration for Io (which gives the highest value among the Galilean satellites), it is reasonable to ignore these effects when investigating the qualitive character of stationary orbits on short time scales. To verify the validity of these solutions, we use the numerical integration method to see the effect of the Galilean moon Io in 800 Jovian days. The orbital elements ()aei,,,Ω ,ω , M of Io that we adopt here are taken from the 10th China Trajectory Optimization Competition, i.e., ()422029.687km, 0. 004308, 0.04 deg,− 79.64 deg,37. 9 91deg, 4.818 deg . Calculation results show that the drift of longitude ( Δλ ) for the satellite on stationary orbits is less than 0.1

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In the following, we show that the effects of third bodies, the Jovian ring and the magnetic ﬁeld of Jupiter, are negligible compared to those of the J2 and J4 terms. Based on the data of the Sun and moons around Jupiter (see for example, [30]), one can calculate the maximal ratio of the disturbed acceleration (ad) and central gravity acceleration (am) for satellites on stationary orbits, which is achieved when the satellite, Jupiter, and the third body are in a straight line. Therefore, the maximal ratio can be calculated as [31] a m r 3 d = 2 0 , (10) am max MJ ρ where m is the mass of the third body, and ρ is the distance between Jupiter and the third body. Values of the maximum ratio (109) for different bodies are reported in Table2. Since the J4 term of Jupiter is 100 times bigger than the ratio of acceleration for Io (which gives the highest value among the Galilean satellites), it is reasonable to ignore these effects when investigating the qualitive character of stationary orbits on short time scales. To verify the validity of these solutions, we use the numerical integration method to see the effect of the Galilean moon Io in 800 Jovian days. The orbital elements (a, e, i, Ω, ω, M) of Io that we adopt here are taken from the 10th China Trajectory Optimization Competi- tion, i.e., 422, 029.687 km , 0.004308, 0.04 deg, −79.64 deg, 37.991 deg, 4.818 deg. Cal- culation results show that the drift of longitude (∆λ) for the satellite on stationary orbits is less than 0.1 deg in 800 Jovian days. The inclination changes no more than 1.0 × 10−4 deg. The oscillation of the semi-major axis is less than 0.13% of the obital radius. However, it should be noted that the obtained solutions may not be correct on long time scales.

Table 2. The disturbing acceleration due to the third body.

−5 −6 The Third Body m/MJ(×10 ) ρ/r0 (ad/am)max(×10 ) Sun ([32]) 1.0473 × 108 4856.8 0.01823 Io ([30,32]) 4.7047 2.6311 5.20000 Europa ([30,32]) 2.5283 4.1868 0.68904 Ganymede ([30,32]) 7.8056 6.6775 0.52379 Callisto ([30,32]) 5.6673 11.7511 0.06989

The Jovian main ring is about 6440 km wide and probably less than 30 km thick. We denote this width by w. Moreover, the distance, d, of the ring from the center of mass of 13 Jupiter is about 122,500 km, and the mass, mr, is about 1.0×10 kg [32]. For satellites on stationary orbits with position rs, the gravitational acceleration due to the Jovian main ring can be calculated as

ZZZ (r(P) − r )ρ(P)dV = s Fr G 3 , (11) ring |r(P) − rs|

where r(P), ρ(P) denote the position vector and the density, respectively, which depend on the position of a point, P, belonging to the ring. We can see that

Gmr |r(P) − rs|≥ r0 − d − w ≈ 160, 240 − 122, 500 − 6440 ≈ 0.4378R, |Fr| ≤ . (0.4378R)2

13 2 Gmr GMJ mr r0 2 1.0×10 2.2414 −13 Note that 2 / 2 = · ≈ 27 · ≈ 1.3795 × 10 , (0.4378R) r0 MJ 0.4378R 1.9×10 0.4378 which is far smaller than J2 and J4 terms. Therefore, the effects of the Jovian main ring on stationary orbits can be neglected. Another perturbation that may affect stationary orbits is the magnetic ﬁeld of Jupiter when the satellite is charged. Here, we brieﬂy analyze the effects under some assumptions. Aerospace 2021, 8, 183 8 of 15

The magnetic ﬁeld of Jupiter near stationary orbits when exterior terms are neglected can be written as B(r, φ, λ) = −∇V(r, φ, λ), (12) where ∞ l l+1 R m V = R∑ ∑ [gl,m cos(mλ) + hl,m sin(mλ)]Pl (cos φ). (13) l=1 m=1 r

Here, gl,m and hl,m are the geomagnetic Gauss coefﬁcients (their values can be found m in [33,34]), and Pl (cos φ) is the normalized Legendre function. The acceleration due to the Lorentz force is q . F = r − ω × r × B(r, φ, λ), (14) L m −12 −1 where q = 4πε0Φs,ε0 = 8.854187817 × 10 F · m is the vacuum permittivity, Φ is the surface potential of satellites, s is the radius of the satellite, ω is the angular velocity vector of Jupiter. For a charged satellite with Φ = 500 V (an in-depth study of the effect of spacecraft charging at Jupiter can be found in [35]), s = 10 m, and m = 100 kg, we have −7 −8 q = 5.5633 × 10 C and FL ≤ 1.14 × 10 N. The Lorentz force compared with the lowest order of gravity provides the ratio

3 |F | n − nJ Bqr L = ≈ 2.3 × 10−11 |G0| GMm

This ratio is also far smaller than J2 and J4. Therefore, it can be concluded that the effects of Lorentz forces on a satellite can be neglected.

3. Sun-Synchronous Orbits The oblate nature of the primary body can lead to a secular variation of the ascending node of the orbit. However, we can use the orbit perturbations to keep the orientation of the Sun line direction ﬁxed with respect to the orbital plane (for example, perpendicular to it) during one revolution of Jupiter around the Sun. Based on the mean element theory, the mean nodal precession rate coming from the secular perturbations of the ﬁrst and second order [36,37] can be described as . 2 9nJ2R4 n 2 √ 2 √ Ω = − 3nJ2R cos i − 2 cos i 3 + e + 1 − e2 − sin2 i 5 − 5e + 3 1 − e2 2a2(1−e2)2 4p4 2 6 3 24 2 h 2 2 i (15) 35J4 6 9e 2 3 9e − 2 7 + 7 − sin i 2 + 4 . 18J2 For sun-synchronous orbits, the mean nodal precession rate is equal to the mean motion of Jupiter orbiting around the Sun (ns). Thus, we get the following equation:

3 g(cos i) = Γ1 cos i + Γ2 cos i + Γ3 = 0, (16)

where " # 9nJ2R4 5 5 3 p 35J 1 3 = 2 − 2 + − 2 − 4 + 2 Γ1 4 e 1 e 2 e , (17) 4p 3 24 2 6J2 2 4 ( " #) 3nJ R2 3J R2 1 3e2 1 p 5J 1 3e2 = 2 + 2 − + − − 2 + 4 + Γ2 2 1 2 1 e 2 , (18) 2p 2p 6 8 2 2J2 2 4

Γ3 = ns. (19) Here, we remark that in Equations (17) and (18), p is equal to a(1−e2). First, setting x = cos i, we note that the equation g(x) = 0 has three real roots if the discriminant satisﬁes the following inequality: ! Γ2 Γ3 = − 3 + 2 ≥ ∆ 2 3 0. (20) 4Γ1 27Γ1 Aerospace 2021, 8, 183 9 of 15

Then, it is also necessary to avoid impact with Jupiter. However, the surface of Jupiter cannot be unambiguously deﬁned since it is a gas giant. The critical perijovian distance, Aerospace 2021, 8, x FOR PEER REVIEW 10 of 17 dc, is usually much larger than the radius of Jupiter, R, due to radiation safety issues. However, let us take for convenience dc = R, so that the semi-major axis and eccentricity have to satisfy The variation of the discriminant dwithc = R respect< a(1 −to e)a. and e is shown in Figure 4a. (21) R We see thatThe the variation discriminants of the are discriminant always negative with respectfor aRR∈[,2] to a and ande is e shown∈−[0,1 in ] Figure, there-4a. a R We see that the discriminants are always negative for a ∈ [R, 2R] and e ∈ [0, 1 − a ], fore,therefore, it can be itconcluded can be concluded that there that usually there usuallyexists one exists meaningful one meaningful inclination inclination for sun-syn- for sun- chronoussynchronous orbits when orbits the when semi-major the semi-major axis and axis eccentricity and eccentricity are given. are given.The inclinations The inclinations for differentfor different semi-major semi-major axis and axis eccentricities and eccentricities are shown are in shown Figure in4b. Figure It was4 b.shown It was that shown the inclinationthat the inclinationmonotonically monotonically increases with increases respect with to respectthe semi-major to the semi-major axis. For axis.the same For the semi-majorsame semi-major axis, the inclination axis, the inclination decreases decreasesas the eccentricity as the eccentricity increases. increases.Moreover, Moreover,the in- clinationsthe inclinations with the withsame thealtitude same ratio, altitude aR/ ratio,, area relatively/R, are relatively smaller than smaller those than of thosenear- of Earthnear-Earth sun-synchronous sun-synchronous orbits. orbits.

(a) (b)

FigureFigure 4. (a 4.) Variation(a) Variation of the of discriminant the discriminant with with respect respect to toa aandand ee;(; (bb)) variationsvariations ofof thethe inclinationinclination i as i asthe the semi-major semi-major axis axis a increasesa increases from fromR toR 2 Rto. 2R .

ForFor satellites satellites on onsun-synchronous sun-synchronous orbits orbits with with medium medium altitude, altitude, perturbations perturbations from from the thegravity gravity of the of third the third body body are usually are usually smaller smaller compared compared to satellites to satellites on stationary on stationary orbits.orbits. However, However, the risk the coming risk comingfrom the from Jupiter the rings Jupiter and ringsthe magnetic and the field magnetic may increase ﬁeld may greatly.increase greatly. ForFor sun-synchronous sun-synchronous orbits, orbits, when when aRa===1.43731.4373R 102755.451= 102, 755.451km andkm ande = e0.4=,0.4 we, we havehave i =i90.183= 90.183deg; deg; when when aR==a 1.5308= 1.5308R 109439.953= 109, 439.953km and km ande = 0.1e =, 0.1,the thecorresponding corresponding Ω ΔΩ inclinationinclination is 90.321 is 90.321 deg. deg. The evolution The evolution of of andΩ andof the of difference the difference between between ∆Ωandand nts nst overover25T 25forT thesefor these two two cases cases are arepresented presented in Figure in Figure 5. In5. Inthe the first ﬁrst case, case, this this difference difference is is aboutabout 0.018 0.018 deg; deg; and and in the in thesecond second case, case, it is it about is about 0.008 0.008 deg. deg.

(a) (b)

Aerospace 2021, 8, x FOR PEER REVIEW 10 of 17

The variation of the discriminant with respect to a and e is shown in Figure 4a. R We see that the discriminants are always negative for aRR∈[,2] and e∈−[0,1 ] , there- a fore, it can be concluded that there usually exists one meaningful inclination for sun-synchronous orbits when the semi-major axis and eccentricity are given. The inclinations for different semi-major axis and eccentricities are shown in Figure 4b. It was shown that the inclination monotonically increases with respect to the semi-major axis. For the same semi-major axis, the inclination decreases as the eccentricity increases. Moreover, the inclinations with the same altitude ratio, aR/ , are relatively smaller than those of near- Earth sun-synchronous orbits.

(a) (b) Figure 4. (a) Variation of the discriminant with respect to a and e ; (b) variations of the inclination i as the semi-major axis a increases from R to 2R .

For satellites on sun-synchronous orbits with medium altitude, perturbations from the gravity of the third body are usually smaller compared to satellites on stationary orbits. However, the risk coming from the Jupiter rings and the magnetic field may increase greatly. For sun-synchronous orbits, when aR==1.4373 102755.451 km and e = 0.4 , we have i = 90.183 deg; when aR==1.5308 109439.953 km and e = 0.1 , the corresponding Ω ΔΩ inclination is 90.321 deg. The evolution of and of the difference between and nts

Aerospace 2021, 8, 183 over 25T for these two cases are presented in Figure 5. In the first case, this difference10 of is 15 about 0.018 deg; and in the second case, it is about 0.008 deg.

Aerospace 2021, 8, x FOR PEER REVIEW 11 of 17

(a) (b)

(c) (d) Ω ΔΩ = Ω − Ω FigureFigure 5. 5.EvolutionEvolution of of Ω andand difference difference between between ∆Ω = Ω − Ω0 0andand ntnsst overover 2525TT, where, whereT isT a Jovian day, Ω =Ω=60 deg is the initial condition (the blue curve was indeed obtained by numerical is a Jovian day,0 0 60 deg is the initial condition (the blue curve was indeed obtained by nu- mericalintegration integration of osculating of osculating elements). elements). (a) Evolution (a) Evolution of Ω offor Ω a for= 102755.451a = 102755.451km, i km,= 90.183 e = 0.4deg., (=b) Difference between ∆Ω and nst for a =ΔΩ102, 755.451 km, e = 0.4,=i = 90.183 deg. (c) Evolution= i 90.183 deg. (b) Difference between and nts for a 102755.451 km, e 0.4 , i of= 90.183 for a = deg.109, (c 439.953) Evolutionkm, ofe =Ω0.1, for i a==90.321109439.953deg. (km,d) Difference e = 0.1, i between= 90.321∆Ω deg.and (d)n sDif-t for a = 109, 439.953ΔΩ km, e = 0.1, i = 90.321= deg. = = ference between and nts for a 109439.953 km, e 0.1, i 90.321 deg. 4. Orbits with Critical Inclination 4. OrbitsThe with variations Critical of Inclination the eccentricity and the argument of perijove are mainly caused byThe the equatorialvariations of bulge the eccentricity of Jupiter. and These the ar usuallygument produce of perijove negative are mainly effects caused on space by themissions equatorial to Jupiter.bulge of However,Jupiter. These we canusually choose produce orbits negative with critical effects inclinations on space missions to avoid tothese Jupiter. disadvantages. However, we can choose orbits with critical inclinations to avoid these disadvantages.According to the mean element theory, the mean variation rate of ω caused by pertur- bationsAccording of the ﬁrstto the and mean second element order theory, [36,37] canthe mean be formulated variation as rate follows: ofω caused by per- . 2 turbations of the9 nJfirst2R4 andn second2 order√ [36,37] can be formulated2 √ as follows: = − 3nJ2R 5 2 i − + 2 + 7e + − e2 − 2 i 103 + 3e + 11 − e2 ω 2a2(1−e2) 2 sin 2 p4 4 12 2 1 sin 12 8 2 1 22422 √ h 2 2 2 i (22) 4 215 15 392nJ22 R15 52 222235J4 12 nJ R27e 7ee2 93 27e  1034 21 381e 11 + sin i ω =−− e + 1 −sine −ieie −+ 22 + 4− +sin i + 2 1+ −− sin+ sin i ++ + 1 −. 48 32 224 18J 7 4 14 14 4 4 16 21ae()− 21212822 p   . (22) To keep the invariance of the mean argument22 of perijove, we set ω = 0. Therefore, 2 we 422215 15 1535J4  12 27ee 2 93 27 4 21 81 e +sinieeobtain −+−−+−+++ the following 1 equation  sin i sin i  . 48 32 42 7 14 14 4 4 16 18J 2  4 2 Γ4 sin i + Γ5 sin i + Γ6 = 0, (23) To keep the invariance of the mean argument of perijove, we set ω = 0. Therefore, wewhere obtain the following equation 2 4 " 2 2 # 9nJ R 215 15e4215 p 35J 21 81e = 2 Γ+Γ+Γ=− sinii+ sin− 2 − 0, 4 + (23) Γ4 4 4561 e 2 , (24) 4p 48 32 4 18J2 4 16 where

935nJ24 R215 15ee22 15 J  21 81 Γ=24 − +1, −e2 − + 4 42 (24) 418pJ48 32 42  4 16

33nJ R22 J R103 3ee22 11 35 J 31 9 Γ=22 −−51, + + −e2 + 4 + 5 22 2 (25) 46pp12 8 2 J2 14 4

33nJ R22 J R22 5 J  Γ=2  +2 +79ee + −2 −4 + 6 44214.e  (26) 22122 2 46pp J2  Note that the left hand side of equation (23) is a second-degree polynomial with respect to sin2 i . This polynomial in sin2 i may have two real roots in [0,1], and so we

Aerospace 2021, 8, 183 11 of 15

" # 3nJ R2 3J R2 103 3e2 11 p 35J 31 9e2 = 2 − − 2 + + − 2 + 4 + Γ5 2 5 2 1 e 2 , (25) 4p p 12 8 2 6J2 14 4 Aerospace 2021, 8, x FOR PEER REVIEW ( " #) 12 of 17 3nJ R2 3J R2 7e2 p 5J 9e2 = 2 + 2 + + − 2 − 4 + Γ6 2 4 2 4 2 1 e 2 4 . (26) 4p p 12 6J2 2 may haveNote up to that four the values left hand of the side inclination of Equation in [0, (23)π ] isthat a second-degreesolve Equation polynomial(23). Let us set with respect to sin2 i. This polynomial in sin2 i may have two real roots in [0,1], and so we x = sin2 i and write Equation (23 as may have up to four values of the inclination in [0, π] that solve Equation (23). Let us set 2 =Γ2 +Γ +Γ = x = sin i and write Equation (23)hx() as 456 x x 0. (27)

Γ> Γ< Γ> 2 Noting that 4560, 0,h(x 0), = weΓ4 findx + thatΓ5x +hx()Γ6 = can0. have up to two real roots. (27) Moreover, if h(1)> 0 , they will fall in the interval [0,1]. If there exist four roots in [0,π ] Noting that Γ > 0, Γ < 0, Γ > 0, we ﬁnd that h(x) can have up to two real roots. of the critical inclination,4 then5 equation6 (27) must have two different real roots lying in Moreover, if h(1) > 0, they will fall in the interval [0,1]. If there exist four roots in [0, π] [0,1]. Observing that Γ>0, Γ< 0, Γ> 0 , we can easily see a necessary condition for of the critical inclination,456 then Equation (27) must have two different real roots lying in Equation (27) to have two different roots lying in [0, 1] is [0,1]. Observing that Γ4 > 0, Γ5 < 0, Γ6 > 0, we can easily see a necessary condition for Equation (27) to have two different=Γ > roots lying =Γ in +Γ [0, 1] +Γ is > hh(0)6456 0, (1) 0 (28)

Moreover, the conditionh( (21)0) = shouldΓ6 > 0, alsoh( be1) =satisfiedΓ4 + Γ to5 + avoidΓ6 > the0 impact with Jupiter. (28) The variation of h(1) for aRR∈[1.1 , 9 ] and eRa∈−[0,1 / ] are presented in Figure 6a. We noteMoreover, that h(1)< the 0 condition in this domain (21) should of ()ae, also, therefore, be satisﬁed there to avoid exist thetwo impact critical with inclina- Jupiter. The variation of h(1) for a ∈ [1.1R, 9R] and e ∈ [0, 1 − R/a] are presented in Figure6a. We tions,note and that oneh( of1) them< 0 in corresponds this domain to of a (retrogradea, e), therefore, orbit. there The variation exist two of critical critical inclinations, inclina- R 2R tionsand of onedirect of orbits them correspondsas a increases to a from retrograde to orbit. are The presented variation in of Figure critical 6b inclinations for differ- of < entdirect values orbits of the as eccentricity.a increases fromWe canR tosee2 Rthatare when presented e is insmall, Figure for6 bexample for different e 0.2 values, the of criticalthe eccentricity.inclination monotonically We can see that decreases when e is with small, respect for example to the altitudee < 0.2, of the the critical orbit. inclination On the > othermonotonically hand, when decreasese 0.4 , the with critical respect inclination to the altitude monotonically of the orbit. increases. On the other It can hand, also whenbe concludede > 0.4 ,that the there critical exist inclination some critical monotonically values for increases. e, such that It canthe critical also be inclination concluded is that independentthere exist of some the criticalaltitude values of the fororbits. e, such that the critical inclination is independent of the altitude of the orbits.

(a) (b)

FigureFigure 6. (a 6.) Variation(a) Variation of h(1) of ash( 1a) functionas a function of the semi-major of the semi-major axis a axisand thea and eccentricity the eccentricity e . (b) e. Variation(b) Variation of the criticalof the inclination critical inclination i as a functioni as a function of the semi-major of the semi-major axis for different axis for differentvalues of valuesthe eccentricity.of the eccentricity.

FigureFigure 6b 6showsb shows that that in inthe the case case of ofJupiter, Jupiter, critical critical inclinations inclinations depend depend on onthe the semi- semi- major axis and eccentricity, while for the Earth considering only the J , they are fixed major axis and eccentricity, while for the Earth considering only the J 2 , they2 are fixed (63.435(63.435 deg deg and and 116.565 116.565 deg). deg). The Theevolution evolution of ω of andω and e fore for two two different different values values of the of the semi-majorsemi-major axis axis and and eccentricity eccentricity are are shown shown in Figures in Figures 7 and7 and 8.8 . For the first case in Figure7, the amplitude of ω is about 5 deg over a time of 25T. The amplitude of e is about 0.013. For the second case, the amplitude of ω is about 1.8 deg over the same time. One can also see from Figure8 that the amplitude of ω in the first case is

Aerospace 2021, 8, 183 12 of 15

AerospaceAerospace 2021 2021, 8,, 8x, FORx FOR PEER PEER REVIEW REVIEW 1313 of of 17 17

much larger. This is reasonable, since the semi-major axis of the second case is larger than that of the ﬁrst case.

(a2) (a1(a1) ) (a2)

(b1(b1) ) (b2(b2) ) ω == = FigureFigure 7. 7.Evolution EvolutionEvolution of of of ω andand and e over eover over25 25 T25,(TTa1, (,,a1 a2(a1,)a2,aa2) =)aR 1.6832aR==1.68321.6832R = 120, 120335.334 120335.334 335.334 km, km, km,i = i 63.497i =63.49763.497deg; ======deg;(deg;b1 ,(b2b1 (b1),b2a,b2=) )2.1488aR aR2.14882.1488R = 153, 153622.009622.009 153622.009 km, i km,= km,63.431 i i 63.431 deg,63.431 where deg, deg, Twhere iswhere a Jovian T T is day. isa aJovian Jovian day. day.

FigureFigure 8. 8.e e~eω~∼ω ωevolution evolutionevolution over over over 25 25TT, ,red, red red points points points correspond correspondcorrespond to toto aR aaR=====1.68321.68321.6832R = 120335.334120, 120335.334 335.334kmkmkm and andand i i=i==63.49763.497deg; deg;deg; green green green points points points correspond correspond correspond to to to aaR =aR====2.14882.1488R = 153, 153622.009 153622.009 622.009 kmkmkm andand andi =i =i63.431=63.43163.431 deg.deg.deg.

5. RepeatingForFor the the first first Ground case case in in TrackFigure Figure Orbits 7, 7, the the amplitude amplitude of of ω ω is is about about 5 5deg deg over over a atime time of of 25 25T T. . TheThe amplitude amplitudeRepeating of groundof e e is is about trackabout 0.013. orbits 0.013. For are For the usually the second second used case, case, for the remote the amplitude amplitude sensing of satellitesof ω ω is is about about to obtain 1.8 1.8 degadeg good over over appearancethe the same same time. time. for orbitalOne One can can covering. also also s eesee Forfrom from satellites Figure Figure 8on 8that that these the the orbits,amplitude amplitude the groundof of ω ω in tracksin the the firstcanfirst case repeatcase is is much periodically.much larger. larger. This This In thisis is reasonable, reasonable, way, the areasince sinceof the the Jupiter semi-major semi-major covered axis axis by of of thethe the satellitesecond second case cancase is be is largerdynamicallylarger than than that that monitored of of the the first first to case. detectcase. the change of the target during this period of time. For

5.5. Repeating Repeating Ground Ground Track Track Orbits Orbits RepeatingRepeating ground ground track track orbits orbits are are usually usually used used for for remote remote sensing sensing satellites satellites to to obtain obtain a agood good appearance appearance for for orbital orbital covering. covering. For For sa satellitestellites on on these these orbi orbits,ts, the the ground ground tracks tracks

Aerospace 2021, 8, 183 13 of 15

example, we can make use of these orbits to detect the change of the Great Red Spot. The interval of adjacent ground track in the equator is . ∆λ = TN nJ − Ω (29)

2π where TN = . . is the nodal period of the motion of the spacecraft. Using the mean M+ω element theory, we have [36,37]

. 2 √ 9nJ2R4 √ n 2√ M = n + 3nJ2R 1 − 3 sin2 i 1 − e2 + 2 1 − e2 1 1 − 3 sin2 i 1 − e2 2p2 2 4p4 2 2 5 10e2 2 19 26e2 4 233 103e2 + 2 + 3 − sin i 3 + 3 + sin i 48 + 12 (30) 4 e 35 35 2 315 4 35J4 2 9 45 2 45 4 + − 2 12 − 4 sin i + 32 sin i − 2 e 14 − 14 sin i + 16 sin i , 1 e 18J2

where n is the mean angular velocity. The conditions for repeating ground track orbits can be formulated as . DTN nJ − Ω = D∆λ = 2πN (31)

where D and N are positive integers. Equation (31) implies satellites will have a repeating ground track after completing D revolutions in N Jovian days. The ground track repetition D parameter, Q = N , which represents the number of orbital resolutions in a Jovian day, is widely used in engineering practice. Using this parameter combined with the definition of TN, Equation (31) is equivalent to . . M + ω Q = . . (32) nJ − Ω We can see that once Q is fixed, a, e and i will be related by Equation (32). The repeating ground track orbits, which are also sun-synchronous, may play an important role in planetary exploration. For these orbits, Equation (16) was also satisfied. The change of inclinations of sun-synchronous repeating ground track orbits, with respect to the semi- major axis for different values of Q, are presented in Figure9. One can see that for a Aerospace 2021, 8, x FOR PEER REVIEW 15 of 17 given Q, small variations of the semi-major axis produce huge changes in the inclination. Moreover, for orbits with the same semi-major axis, the inclination,i will be larger if Q is larger. For the Earth and Mars, the inclination is not so sensitive to the semi-major axis [9].

FigureFigure 9.9. InclinationsInclinations ofof sun-synchronoussun-synchronous repeating groundground track orbits as a functionfunction of thethe semi-semi- majormajor axis,axis, forfor differentdifferent valuesvalues ofof thethe repetitionrepetition parameterparameterQ Q. . 6. Conclusions 6. Conclusions In this paper, we analyzed some special orbits around Jupiter considering only the In this paper, we analyzed some special orbits around Jupiter considering only the J2 and J4 terms of the non-spherical gravitational potential. First, stationary orbits were J J investigated2 and 4 in terms spherical of the coordinates. non-spherical The radius gravitational of stationary potential. orbits First, was found stationary to be 2.2414orbits Jupiterwere investigated radii. The longitude in spherical drift coordinates. due to the non-sphericalThe radius of componentstationary orbits of the was gravity found ﬁeld to be 2.2414 Jupiter radii. The longitude drift due to the non-spherical component of the

gravity field did not occur for the lack of J 22 terms. Moreover, a small perturbation of r and φ would lead to the oscillation phenomenon in the radial and North-South directions. We also showed that the perturbations of the Sun and the Galilean satellites, the main ring and the magnetic field are negligible on short time scales compared to those

represented by the terms J 2 and J 4 . However, the obtained solutions may be incorrect on long time scales. Then, sun-synchronous orbits, orbits at critical inclinations, and repeating ground track orbits were discussed based on the mean-element theory. The results showed that only one meaningful inclination exists for sun-synchronous orbits when a and e are fixed in a suitable range. Additionally, only two critical inclinations exist for the critical orbits. Repeating ground track orbits, which are also sun-synchronous, were

also calculated here. The traditional frozen orbits were not discussed because the J3 term was not taken into account in our gravity model. However, we remark that there may exist some non-traditional frozen orbits around Jupiter, which will be left for our future investigations. The dynamics around Jupiter are very complicated and even if some perturbations can be neglected for preliminary orbit design, other effects should be considered. For example, orbits at low-to-moderate inclinations within a few Jovian radii may loiter inside Jupiter’s radiation belts. This can lead to considerable risk for the integrity of a satellite. However, we believe that our work can help to reduce or eliminate the need for station keeping in future space mission around Jupiter.

Author Contributions: Conceptualization, H.Z. and Y.L.; methodology, H.L. and Y.J.; software, Y.J.; validation, H.L., Y.J., and Y.L.; formal analysis, H.L.; investigation, H.Z.; resources, Y.J.; data curation, H.Z. and Y.L.; writing—original draft preparation, Y.L.; writing—review and editing, H.L. and Y.J.; visualization, H.Z. and Y.L.; supervision, H.L. and Y.J.; project administration, H.L. and Y.J.; funding acquisition, Y.J. All authors have read and agreed to the published version of the manuscript. Funding: This research was supported by the National Natural Science Foundation of China (Grant No. 11772356). Acknowledgments: The authors gratefully acknowledge the reviewers for their careful work and thoughtful suggestions that have helped improve this paper substantially.

Aerospace 2021, 8, 183 14 of 15

did not occur for the lack of J22 terms. Moreover, a small perturbation of r and φ would lead to the oscillation phenomenon in the radial and North-South directions. We also showed that the perturbations of the Sun and the Galilean satellites, the main ring and the magnetic ﬁeld are negligible on short time scales compared to those represented by the terms J2 and J4. However, the obtained solutions may be incorrect on long time scales. Then, sun- synchronous orbits, orbits at critical inclinations, and repeating ground track orbits were discussed based on the mean-element theory. The results showed that only one meaningful inclination exists for sun-synchronous orbits when a and e are ﬁxed in a suitable range. Additionally, only two critical inclinations exist for the critical orbits. Repeating ground track orbits, which are also sun-synchronous, were also calculated here. The traditional frozen orbits were not discussed because the J3 term was not taken into account in our gravity model. However, we remark that there may exist some non-traditional frozen orbits around Jupiter, which will be left for our future investigations. The dynamics around Jupiter are very complicated and even if some perturbations can be neglected for preliminary orbit design, other effects should be considered. For example, orbits at low-to-moderate inclinations within a few Jovian radii may loiter inside Jupiter’s radiation belts. This can lead to considerable risk for the integrity of a satellite. However, we believe that our work can help to reduce or eliminate the need for station keeping in future space mission around Jupiter.

Author Contributions: Conceptualization, H.Z. and Y.L.; methodology, H.L. and Y.J.; software, Y.J.; validation, H.L., Y.J. and Y.L.; formal analysis, H.L.; investigation, H.Z.; resources, Y.J.; data curation, H.Z. and Y.L.; writing—original draft preparation, Y.L.; writing—review and editing, H.L. and Y.J.; visualization, H.Z. and Y.L.; supervision, H.L. and Y.J.; project administration, H.L. and Y.J.; funding acquisition, Y.J. All authors have read and agreed to the published version of the manuscript. Funding: This research was supported by the National Natural Science Foundation of China (Grant No. 11772356). Acknowledgments: The authors gratefully acknowledge the reviewers for their careful work and thoughtful suggestions that have helped improve this paper substantially. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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